NIOS Class 12 Question Paper 2021 (Jan Feb) Maths

This Question Paper consists of 33 questions and 16 printed pages + Graph Sheet.
Bg àíZ-nÌ _| 33 àíZ VWm 16 ‘w{ÐV n¥ð> + J«m’$ erQ> h¢&
Sl. No.

Roll No.
AZw H « $ _m§ H $
Code No.
H$mo S > Z§ .
60/OSS/1
MATHEMATICS
Set / go Q > A
(J{UV)
(311)
Day and Date of Examination :
(narjm H$m {XZ d {XZm§ H $)
Signature of Invigilators : 1.
({ZarjH$m| Ho $ hñVmja)
2.
General Instructions :
1. Candidate must write his/her Roll Number on the first page of the Question Paper.
2. Please check the Question Paper to verify that the total pages and total number of questions contained in
the Question Paper are the same as those printed on the top of the first page. Also check to see that the
questions are in sequential order.
3. Making any identification mark in the Answer-Book or writing Roll Number anywhere other than the
specified places will lead to disqualification of the candidate.
4. Write your Question Paper Code No. 60 / OSS /1, Set - A on the Answer-Book.
5. (a) The Question Paper is in English/Hindi medium only. However, if you wish, you can answer in any
one of the languages listed below :
English, Hindi, Urdu, Punjabi, Bengali, Tamil, Malayalam, Kannada, Telugu, Marathi, Oriya, Gujarati,
Konkani, Manipuri, Assamese, Nepali, Kashmiri, Sanskrit and Sindhi.
You are required to indicate the language you have chosen to answer in the box provided in the
Answer-Book.
(b) If you choose to write the answer in the language other than Hindi and English, the responsibility
for any errors/mistakes in understanding the questions will be yours only.
gm_mÝ` AZwXoe :
1. narjmWu àíZ-nÌ Ho$ nhbo n¥ð> na AnZm AZwH«$‘m§H$ Amdí¶ {bI|&
2. H¥$n¶m àíZ-nÌ H$mo Om±M b| H$s àíZ-nÌ Ho$ Hw$b n¥ð>m| VWm àíZm| H$s CVZr hr g§»¶m h¡ {OVZr àW‘ n¥ð> Ho$ g~go D$na N>nr h¡& Bg ~mV
H$s Om±M ^r H$a b| H$s àíZ H«${‘H$ ê$n ‘| h¢&
3. CÎma-nwpñVH$m ‘| nhMmZ-{M• ~ZmZo AWdm {Z{X©ï> ñWmZm| Ho$ A{V[aº$ H$ht ^r AZwH«$‘m§H$ {bIZo na narjmWu H$mo A¶mo½¶ R>ham¶m Om¶oJm&
4. AnZr CÎma-nwpñVH$m na àíZ-nÌ H$r H$moS> g§»¶m. 60 / OSS /1, goQ> - A {bI|&
5. (H$) àíZ-nÌ Ho$db qhXr/A§J«oOr ‘mܶ‘ ‘| h¡& {’$a ^r, ¶{X Amn Mmh| Vmo ZrMo Xr JB© {H$gr EH$ ^mfm ‘| CÎma Xo gH$Vo h¢ : A§J«oOr, qhXr,
CXÿ©, n§Om~r, ~§Jbm, V{‘b, ‘b¶mb‘, H$ÝZ‹S>, VobwJy, ‘amR>r, C{‹S>¶m, JwOamVr, H$m|H$Ur, ‘{Unwar, Ag{‘¶m, Zonmbr, H$í‘rar,
g§ñH¥$V Am¡a qgYr&
H¥$n¶m CÎma-nwpñVH$m ‘| {XE JE ~m°³g ‘| {bI| {H$ Amn {H$g ^mfm ‘| CÎma {bI aho h¢&
(I) ¶{X Amn qhXr Ed§ A§J«oOr Ho$ A{V[a³V {H$gr Aݶ ^mfm ‘| CÎma {bIVo h¢, Vmo àíZm| H$mo g‘PZo ‘| hmoZo dmbr Ìw{Q>¶m|/Jb{V¶m| H$s
{Oå‘oXmar Ho$db AmnH$s hmoJr&

60/OSS/1/311-A] G-607 1 *60/OSS/1/311-A* [ Contd......

MATHEMATICS
(J{UV)
(311)
Time : 3 Hours] [Maximum Marks : 100
g_` : 3 KÊQ>o] [nyUmªH$ : 100
Note : (1) This question paper consists of four Sections A, B, C and D containing 33 questions.

(2) Question Number 1 to 10 in Section A are multiple choice questions (MCQ). Each question carries
one mark. In each question there are four choices (A), (B), (C) and (D) of which only one is correct.
You have to select the correct choice and indicate it in your answer book by writing (A), (B), (C) or (D)
as the case may be. No separate time is allotted for attempting MCQ.

(3) Question Number 11 to 16 in Section B are very short answer questions and carry 2 marks each.

(4) Question Number 17 to 28 in Section C are short answer questions and carry 4 marks each.

(5) Question Number 29 to 33 in Section D are long answer questions and carry 6 marks each.

(6) All questions are compulsory. There is no overall choice, however, alternative choices are given in
some questions. In such questions, you have to attempt only one choice.

{ZX} e : (1) Bg àíZ nÌ ‘| Hw$b 33 àíZ h¢, Omo Mma IÊS>m| A, ~, g VWm X ‘| {d^m{OV h¢&

(2) IÊS>-A ‘| àíZ g§»¶m 1 go 10 VH$ VWm ~hþ{dH$ënr¶ àíZ h¢, {OZ‘| à˶oH$ Ho$ {bE 1 A§H$ {ZYm©[aV h¡& à˶oH$ àíZ Ho$ CÎma
Ho$ ê$n ‘| (A), (B), (C) VWm (D) Mma {dH$ën {XE JE h¢ {OZ ‘| go H$moB© EH$ ghr h¡& AmnH$mo ghr {dH$ën MwZZm h¡ VWm AnZr
nwpñVH$m ‘| (A), (B), (C) VWm (D) ‘| Omo ghr hmo CÎma Ho$ ê$n ‘| {bIZm h¡& ~hþ{dH$ënr¶ àíZ hb H$aZo Ho$ {bE AbJ go g‘¶
Zht {X¶m J¶m h¡&

(3) IÊS> – ~ ‘| àíZ g§»¶m 11 go 16 VH$ A{V bKwCÎmar¶ àíZ h¡ VWm à˶oH$ Ho$ 2 A§H$ {ZYm©[aV h¢&

(4) IÊS >– g ‘| àíZ g§»¶m 17 go 28 VH$ CÎmar¶ àíZ h¡ VWm à˶oH$ Ho$ 4 A§H$ {ZYm©[aV h¢&

(5) IÊS> – X ‘| àíZ g§»¶m 29 go 33 VH$ XrK© CÎmar¶ àíZ h¡ VWm à˶oH$ Ho$ 6 A§H$ {ZYm©[aV h¢&

(6) g^r àíZ A{Zdm¶© h¢& nyU© àíZnÌ ‘| {dH$ën Zht h¢, {’$a ^r Hw$N> àíZmo§ ‘|, Am§V[aH$ {dH$ën h¢& Eogo g^r àíZm| ‘| go AmnH$mo
EH$ hr {dH$ën hb H$aZm h¡&

60/OSS/1/311-A] G-607 2 *60/OSS/1/311-A* [ Contd......

SECTION-A
IÊS-A

 2 1
1. If A 1    , then A is equal to [1]
 1 1 

 2 1  2 1 
(A)   (B)  
 1 1   1 1 

 1 1  2 1
(C)   (D)  
 1 2   1 1

 2 1
¶{X A 1    h¡, Vmo A ~am~a h¡ :
 1 1

 2 1  2 1 
(A)   (B)  
 1 1   1 1 

 1 1  2 1
(C)   (D)  
 1 2   1 1

2. cos(tan–1 x) is equal to [1]

1 1
(A) (B)
x 1
2
1  x2

1
(C) (D) x2  1
x2  1

60/OSS/1/311-A] G-607 3 *60/OSS/1/311-A* [ Contd......

cos(tan–1 x) ~am~a h¡ :

1 1
(A) (B)
x 1
2
1  x2

1
(C) (D) x2  1
x 1
2

3. Let f :    be defined by f (x) = 2x + 3, x. Then f is [1]

(A) one to one but not onto

(B) onto but not one to one

(C) neither one to one nor onto

(D) one to one and onto

‘mZm f :    na f (x) = 2x + 3, x Ûmam n[a^m{fV EH$ ’$bZ h¡& Vmo f EH$
(A) EH¡$H$s naÝVw AmN>mXH$ Zht, ’$bZ h¡

(B) AmN>mXH$ naÝVw EH¡$H$s Zht, ’$bZ h¡

(C) Z Vmo EH¡$H$s Am¡a Z hr AmN>mXH$, ’$bZ h¡

(D) EH¡$H$s Am¡a AmN>mXH$, ’$bZ h¡

3 2 x
4. 2
2 x  x3
dx is equal to [1]

1
(A) (B) 0
2

(C) –1 (D)  4

60/OSS/1/311-A] G-607 4 *60/OSS/1/311-A* [ Contd......

3 2 x
2
2 x  x3
dx ~am~a h¡ :

1
(A) (B) 0
2

(C) –1 (D)  4

 1 1 
5.   log x   log x 2  dx is equal to [1]
 

x 2x
(A) c (B) c
log x log x

x 1
(C) c (D) c
2log x log x

 1 1 
  log x  log x 2  dx ~am~a h¡ :
 
 

x 2x
(A) c (B) c
log x log x

x 1
(C) c (D) c
2log x log x

60/OSS/1/311-A] G-607 5 *60/OSS/1/311-A* [ Contd......

dy
6. If y = xtan x, then is euqal to [1]
dx

tan x  x(log x)sec 2 x tan x  x(log x)sec2 x
(A) y (B) y
x x

(C)
 tan x  sec x  2

(D)
 x tan x  sec x 
2

x x

dy
¶{X y = xtan x h¡, Vmo ~am~a h¡ :
dx

tan x  x(log x)sec 2 x tan x  x(log x)sec2 x
(A) y (B) y
x x

(C)
tan x  sec 2 x
(D)
 x tan x  sec x 
2

x x

dy
7. General solution of the differential equation is  4 tan y equal to [1]
dx

(A) sin y = cex (B) y = sin–1(ce4x)

(C) y = cos–1(ce4x) (D) y = sin–1(ce–4x)

dy
AdH$b g‘rH$aU  4 tan y H$m ì¶mnH$> hb h¡ :
dx

(A) sin y = cex (B) y = sin–1(ce4x)

(C) y = cos–1(ce4x) (D) y = sin–1(ce–4x)

60/OSS/1/311-A] G-607 6 *60/OSS/1/311-A* [ Contd......

8. Which of the following statement is true? [1]

(A) Chord of a circle is double of its radius

(B) Concentric circles have different radius

(C) If a number has more than two factors then it is not composite

(D) 25 is a multiple of 8.

{ZåZ ‘| go H$m¡Zgm H$WZ ghr h¡?

(A) EH$ d¥Îm H$s Ordm CgHo$ AY©-ì¶mg go XþJZr hmoVr h¡&

(B) EH$ hr Ho$ÝÐ dmbo d¥Îmm| H$m AY©-ì¶mg {^ÝZ hmoVo h¢&

(C) ¶{X {H$gr EH$ g§»¶m Ho$ JwUZI§S> Xmo go A{YH$ h¢, Vmo dh g§»¶m ^mÁ¶ Zht h¡&

(D) 25, 8 H$m JwUO h¡&

9. The value of x for which f (x) = |x – 1| is not differentiable, is; [1]

(A) –1 (B) 2

(C) 0 (D) 1

x H$m dh ‘mZ, {OgHo$ {bE f (x) = |x – 1| Ûmam n[a^m{fV ’$bZ AdH$bZr¶ Zht h¡, h¡ :

(A) –1 (B) 2

(C) 0 (D) 1

60/OSS/1/311-A] G-607 7 *60/OSS/1/311-A* [ Contd......

10. If the perpendicular distance from the point (2, K, 0) to the plane 4x – 2y + 3z = 12
4
is unit, then possible values of K are : [1]
29

(A) 1, 4 (B) –1, 3

(C) 0, –4 (D) 0, 4

4
{~ÝXþ (2, K, 0) go Vb 4x – 2y + 3z = 12 H$s bå~dV² Xÿar BH$mB© h¡& K Ho$ gå^dV… ‘mZ h¢ :
29

(A) 1, 4 (B) –1, 3

(C) 0, –4 (D) 0, 4

SECTION-B

IÊS - ~

 0 7 43 
 0 47  , then show that |A| = 0.
11. If A   7 [2]
 43 47 0 


 0 7 43 
¶{X A   7 0 47  h¡, Vmo Xem©BE H$s{OE {H$ |A| = 0
 43 47 0 


OR/AWdm

1 2
, verify that A 2   A  .
2
For A   
3 2

1 2
 
2
A  Ho$ {bE g˶m{nV H$s{OE {H$ A 2
 A
3 2

60/OSS/1/311-A] G-607 8 *60/OSS/1/311-A* [ Contd......

12. Let R be relation defined on the set of natural numbers  as follows R = {(x, y) :
x, y and 3x – y = 12}. Find the domain and range of the relation R. [2]

àmH¥$V g§»¶mAm| Ho$ g‘wƒ¶  na, gå~ÝY R {ZåZ Ûmam n[a^m{fV h¡ :

R = {(x, y) : x, y Am¡a 3x – y = 12} gå~ÝY R H$m àmÝV d n[aga kmV H$s{OE&

tan x  sin x
13. Evaluate lim . [2]
x 0 x3

tan x  sin x
lim H$m ‘mZ kmV H$s{OE&
x 0 x3

14. Find the derivative of sin x3 with respect ot x2. [2]

sin x3 H$m, x2 Ho$ gmnoj, AdH$bO kmV H$s{OE&

 
15. If a  2iˆ  ˆj  3kˆ and b  3iˆ  5 ˆj  2kˆ represent two adjacent sides of a triangle,
then find the angle between them. [2]


¶{X EH$ {Ì^wO H$s Xmo g§b½Z ^wOmE± a  2iˆ  ˆj  3kˆ VWm b  3iˆ  5 ˆj  2kˆ Ûmam {Zê${nV hmo Vmo, BZ
^wOmAm| Ho$ ~rM H$m H$moU kmV H$s{OE&

60/OSS/1/311-A] G-607 9 *60/OSS/1/311-A* [ Contd......

16. Write the converse the following statements : [2]

a) If game is cancelled, then team A is win.

b) If a is a multiple of b then b is a factor of a.

{ZåZ H$WZm| Ho$ {dbmo‘ {b{IE :

a) ¶{X Iob aÔ hmoVm h¡, Vmo Q>r‘ A OrVVr h¡&

b) ¶{X a, b H$m JwUO h¡, Vmo b, a H$m JwUZI§S> h¡&

SECTION - C

IÊS - g

a 2  2a 2a  1 1
a  2 1   a  1
3
2a  1
17. Show that . [4]
3 3 1

a 2  2a 2a  1 1
2a  1 a  2 1   a  1
3
Xem©BE {H$
3 3 1

 3 1
18. If A    , find x and y so that A2 + xI2 = yA. [4]
 6 5

 3 1
¶{X A    Ho$ {bE A2 + xI2 = yA h¡, Vmo x Am¡a y H$m ‘mZ kmV H$s{OE&
 6 5

60/OSS/1/311-A] G-607 10 *60/OSS/1/311-A* [ Contd......

1 6 8 
19. Solve the following equation for x(x > 0) : sin  sin 1  [4]
x x 2

1 6 8 
x(x > 0) Ho$ {bE {ZåZ g‘rH$aU H$mo hb H$s{OE : sin  sin 1 
x x 2

20. Determine the values of a, b for which the function [4]

| x  2 |
 x  2  a , x  2,

f ( x )  a  b , x  2,

2 x  b , x  2


is continuous at x = –2.
a Am¡a b Ho$ do ‘mZ kmV H$s{OE, {OZHo$ {bE ’$bZ

| x  2 |
 x  2  a , x  2,

f ( x )  a  b , x  2,

2 x  b , x  2


x = –2 na gVV h¡&

21. Find the interval in which the function f (x) = 2x3 + 9x2 + 12x + 20 are increasing or
decreasing. [4]

do A§Vamb kmV H$s{OE, {OZ na ’$bZ f (x) = 2x3 + 9x2 + 12x + 20 dY©‘mZ ¶m ömg‘mZ h¡&

OR/AWdm

60/OSS/1/311-A] G-607 11 *60/OSS/1/311-A* [ Contd......

Verify Rolle's theorem for the function f (x) = ex sin x, 0  x  .

’$bZ f (x) = ex sin x, 0  x   Ho${bE amobo Ho$ à‘o¶ H$m g˶mnZ H$s{OE&

22. Divide the number 75 into two parts such that product of one part and square of
the other part is maximum. [4]

g§»¶m 75 H$mo Xmo ^mJm| ‘| {d^m{OV H$s{OE, {Oggo nhbo ^mJ Am¡a Xþgao ^mJ Ho$ dJ© H$m JwUZ’$b A{YH$V‘
h¡&

x2
23. Find  x 2  4 x 2  9 dx .
   [4]

x2
  x  4  x  9  dx kmV H$s{OE&
2 2

OR/AWdm

1
Find :  dx .
x x

1
x x
dx kmV H$s{OE&

 /2 1
24. Evaluate :  0 dx . [4]
1  tan x
 /2 1
 0
1  tan x
dx H$m ‘mZ kmV H$s{OE&

60/OSS/1/311-A] G-607 12 *60/OSS/1/311-A* [ Contd......

25. Solve the following differential equation : [4]

dy 
 sin( x  y )  sin( x  y ), given y  when x = 0.
dx 2

{ZåZ AdH$b g‘rH$aU H$mo hb H$s{OE :

dy
 sin( x  y )  sin( x  y ) ,
dx


{X¶m h¡ {H$ O~ x = 0 Vmo y 
2

OR/AWdm

Form the differential equation corresponding to y = ex(a cosx + b sinx) by eliminating
'a' and 'b'.

a Am¡a b H$m {dbmonZ H$aVo hþE y = ex(a cosx + b sinx) Ho$ g§JV AdH$b g‘rH$aU ~ZmBE&

26. Find the equation of the line passing through the point (–1, –3, –2) and perpendicular
x y z x  2 y 1 z 1
to the lines   and   . [4]
1 2 3 3 2 5

x y z x  2 y 1 z 1
{~ÝXþ (–1, –3, –2) go JwOaZo dmbr Am¡a aoImAm|   VWm   Ho$ bå~dV²
1 2 3 3 2 5
aoIm H$m g‘rH$aU kmV H$s{OE&

60/OSS/1/311-A] G-607 13 *60/OSS/1/311-A* [ Contd......

27. Find the distance of the point (1, 2, 3) from the plane x – y + z = 5 measured
x 1 y  2 z  3
parallel to the line   . [4]
2 3 4

x 1 y  2 z  3
aoIm   Ho$ g‘mÝVa, {~ÝXþ (1, 2, 3) go Vb x – y + z = 5 H$s Xÿar kmV H$s{OE&
2 3 4

28. Let + be the set of all positive real numbers and f : +  [2, ) be function such
that f (x) = x2 + 2. Show that inverse of f exists and also find f –1. [4]

‘mZm YZmË‘H$ dmñV{dH$ g§»¶mAm| H$m g‘wƒ¶ + h¡ Am¡a f : +  [2, ) na f (x) = x2 + 2 Ûmam
n[a^m{fV ’$bZ h¡& Xem©BE {H$ ’$bZ f Ho$ à{Vbmo‘ H$m ApñVËd h¡ Am¡a f –1 ^r kmV H$s{OE&

SECTION-D

IÊS -X

29. Solve the following system of linear equations, using matrix method : [6]

3x + y + z = 1

2x + 2z = 0

5x – y + 2z = 4

Amì¶yh {d{Y go, {ZåZ g‘rH$aU {ZH$m¶ H$mo hb H$s{OE :
3x + y + z = 1

2x + 2z = 0

5x – y + 2z = 4

OR/AWdm

60/OSS/1/311-A] G-607 14 *60/OSS/1/311-A* [ Contd......

Using elementary row operations, find the inverse of the following matrix.

0 1 1
1 2 1
 
 2 3 2
 

àmapå^H$ n§{³V g§{H«$¶mAm| H$m à¶moJ H$aHo$, {ZåZ Amì¶yh H$m ì¶wËH«$‘ kmV H$s{OE :

0 1 1
1 2 1
 
 2 3 2
 

30. Find the equations of the tangent and normal to the curve y (x – 2) (x – 3) – x + 7 = 0 at
the point where it cuts the x axis. [6]

dH«$ y (x – 2) (x – 3) – x + 7 = 0 Ho$ Cg {~ÝXþ, Ohm± ¶h dH«$ x-Aj H$mo H$mQ>Vr h¡, na ñne© aoIm d
A{^b§~ Ho$ g‘rH$aU kmV H$s{OE&

31. Find the smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2. [6]

d¥Îm x2 + y2 = 4 VWm aoIm x + y = 2 Ûmam {Kao N>moQ>o joÌ H$m joÌ’$b kmV H$s{OE&

OR/AWdm

Find the derivative of tan x from first principle.

tan x H$m àW‘ {gÕmÝV go AdH$bZ kmV H$s{OE&

60/OSS/1/311-A] G-607 15 *60/OSS/1/311-A* [ Contd......

32. Find the value of   for which the four points A,B,C,D with position vectors
 ˆj  kˆ, 4iˆ  5 ˆj   kˆ, 3iˆ  9 ˆj  4kˆ, 4iˆ  4 ˆj  4kˆ respectively are coplanar. [6]

 H$m dh ‘mZ kmV H$s{OE {OgHo$ {bE Mma {~ÝXþ A,B,C,D {OZH$s pñW{V g{Xe H«$‘e…  ˆj  kˆ,
4iˆ  5 ˆj   kˆ, 3iˆ  9 ˆj  4kˆ, 4iˆ  4 ˆj  4kˆ ghVbr¶ hm|&

33. A manufacutrer makes red and blue pen. He works at least 10 hours per day. A red
pen takes twice as much as time as to make a blue pen. A blue pen takes 20 minutes
time to make. A red pen sells for K8 and at most 50 can be sold in a day. A blue pen
sells for K5 and at most 50 can be sold in a day. The manufacturer desires to
maximize his revenue. Formulate the above problem as a L.P.P. and solve it
grpahically. [6]

EH$ {Z‘m©Vm bmb VWm Zrbo n¡Z ~ZmVm h¡& dh à{V{XZ H$‘ go H$‘ 10 K§Q>o H$m‘ H$aVm h¡& EH$ bmb n¡Z ~ZmZo
‘|, EH$ Zrbo n¡Z H$mo ~ZmZo ‘| bJZo dmbo g‘¶ go XþJZm g‘¶ bJVm h¡& EH$ Zrbo n¡Z H$mo ~ZmZo ‘| 20 {‘ZQ>
bJVo h¢& EH$ bmb n¡Z H$mo K8 ‘| ~oMm OmVm h¡ Am¡a EH$ {XZ ‘| A{YH$V‘ 50 ZJ ~oMo Om gH$Vo h¢& EH$ Zrbm
n¡Z K5 ‘| ~oMm OmVm h¡ Am¡a EH$ {XZ ‘| A{YH$V‘ 50 ZJ ~oM| Om gH$Vo h¢& {Z‘m©Vm A{YH$V‘ amOñd A{O©V
H$aZm MmhVm h¡& Bg g‘ñ¶m H$mo a¡{IH$ àmoJ«m‘Z g‘ñ¶m ~ZmBE Am¡a AmboIr¶ {d{Y go hb H$s{OE&

  

60/OSS/1/311-A] G-607 16 *60/OSS/1/311-A*

New Questions of SET - B

SECTION - A
5. Distance between planes x + 2y – 3z = 10 and 2x + 4y – 6z = –4 is
14
(A) (B) 7
2
14 12
(C) (D)
12 14
Vbm| x + 2y – 3z = 10 VWm 2x + 4y – 6z = –4 Ho$ ~rM H$s Xÿar h¡ :
14
(A) (B) 7
2
14 12
(C) (D)
12 14

SECTION - C
 4 3 3
17. I3 is an identity matrix of order 3 and A   1 0 1  . Show that A2 = I3 and
 4 4 3 
 
hence find A–1 and (A2)–1.
 4 3 3
H$mo{Q> 3 H$m EH$ BH$mB© Amì¶yh I3 VWm A   1 0 1  Ho$ {bE Xem©BE {H$ A2 = I3 VWm A–1 Am¡a
 4 4 3 
 
(A2)–1 ^r kmV H$s{OE&

19. The perimeter of a triangle is 16cm. If one side is 6cm, find the other two sides so
that the area of the triangle is maximum.
EH$ {Ì^wO H$m n[a‘mn 16 go‘r h¡& ¶{X EH$ ^wOm 6 go‘r bå~r hmo, Vmo ~mH$s Xmo ^yOmAm| H$s bå~mB©¶m± kmV
H$s{OE, {Oggo {Ì^wO H$m joÌ’$b A{YH$V‘ hmo&

60/OSS/1/311-A] G-607 17 *60/OSS/1/311-A*

27. Solve the following differential equation :
dy
x cos x  y ( x sin x  cos x)  1
dx
dy
AdH$b g‘rH$aU x cos x  y ( x sin x  cos x)  1 H$mo hb H$s{OE&
dx
OR/AWdm

dy
Solve the differential equation : ( x  2)  x2  4x  5 .
dx
dy
AdH$b g‘rH$aU ( x  2)  x 2  4 x  5 H$mo hb H$s{OE&
dx

SECTION - D
29. Find the equation of the normals to the curve x2 + y2 – 2x – 4y + 1 = 0 at the points
where tangent are parallel to the y-axis.
dH«$ x2 + y2 – 2x – 4y + 1 = 0 Ho$ CZ {~ÝXþAm|, {OZ na dH«$ na S>mbr JB© ñne© aoIm y-Aj Ho$ g‘mÝVa
h¡, na A{^b~m| H$m g‘rH$aU kmV H$s{OE&

30. Find the image of the point (1, 3, 4) on the plane 2x – y + z = 9.
Vb 2x – y + z = 9 ‘| {~ÝXþ (1, 3, 4) H$m à{V{~å~ kmV H$s{OE&

60/OSS/1/311-A] G-607 18 *60/OSS/1/311-A*

New Questions of SET - C

SECTION - A
5. Distance between planes x + 2y – 3z = 10 and 2x + 4y – 6z = 4 is

14
(A) (B) 8
2

14 8
(C) (D)
8 14
Vbm| x + 2y – 3z = 10 VWm 2x + 4y – 6z = 4 Ho$ ~rM H$s Xÿar h¡ :
14
(A) (B) 8
2

14 8
(C) (D)
8 14

SECTION - C
23. Solve the following differential equation :
dy
 y cot x  2cos x
dx
dy
AdH$b g‘rH$aU  y cot x  2cos x H$mo hb H$s{OE&
dx
OR/AWdm
Find the equation of the curve represented by
dy
 xy  x  y  1
dx
and passing through the point (2, 0).
dy
 xy  x  y  1 Ûmam àX{e©V dH«$, Omo {~ÝXþ (2, 0) go hmoH$a OmVm h¡, H$m g‘rH$aU kmV H$s{OE&
dx

60/OSS/1/311-A] G-607 19 *60/OSS/1/311-A*

25. Given the sum of the perimeters of a circle and square, show that the sum of their
areas is least when the diameter of the circle is equal to side of the square.
{XE JE d¥Îm H$s n[a{Y VWm dJ© Ho$ n[a‘mn H$m ¶moJ Ho$ {bE, Xem©BE {H$ BZHo$ joÌ’$bm| H$m ¶moJ ݶyZV‘ hmoJm
O~{H$ d¥Îm H$m ì¶mg, dJ© H$s ^wOm Ho$ ~am~a hmo&

 5 8 0 
 
27. I3 is an identity matrix of order 3 and A   3 5 0  . Show that A2 = I3 and
 1 2 1
 
hence find A–1 and (A2)–1.
 5 8 0 
H$mo{Q> 3 H$m EH$ BH$mB© Amì¶yh I3 VWm A   3 5 0  Ho$ {bE Xem©BE {H$ A2 = I3 VWm A–1 Am¡a
 1 2 1
 
(A2)–1 ^r kmV H$s{OE&

SECTION - D
x 1 y  2 z  3 x 5 y 8 z 6
29. Prove that the straight lines   and   are
3 1 2 7 5 11
coplanar and find the equations of plane on which they lie.
{gÕ H$s{OE {H$ aoImE±
x 1 y  2 z  3 x  5 y  8 z  6
  ;  
3 1 2 7 5 11
g‘Vbr¶ h¢& Cg Vb H$m g‘rH$aU ^r kmV H$s{OE {Og‘| ¶o aoImE± pñWV h¢&

33. Find the equations of the tangent and normal to the curve y2 = x3 at the point where
x coordinate is 4.
dH«$ y2 = x3 Ho$ Cg {~ÝXþ na {OgH$m x-{ZX}em§H$ 4 h¡, ñne© aoIm d A{^b§~ Ho$ g‘rH$aU kmV H$s{OE&

60/OSS/1/311-A] G-607 20 *60/OSS/1/311-A*